1.02j Manipulate polynomials: expanding, factorising, division, factor theorem

1

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 7 x - 3\).
   1. Find \(\mathrm { f } ( - 1 )\).
   2. Use the Factor Theorem to show that \(2 x + 1\) is a factor of \(\mathrm { f } ( x )\).
   3. Simplify the algebraic fraction \(\frac { 4 x ^ { 3 } - 7 x - 3 } { 2 x ^ { 2 } + 3 x + 1 }\).
2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 4 x ^ { 3 } - 7 x + d\). When \(\mathrm { g } ( x )\) is divided by \(2 x + 1\), the remainder is 2 . Find the value of \(d\).

1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 15 x ^ { 3 } + 19 x ^ { 2 } - 4\).

1. 1. Find \(\mathrm { f } ( - 1 )\).
   2. Show that \(( 5 x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
2. Simplify $$\frac { 15 x ^ { 2 } - 6 x } { f ( x ) }$$ giving your answer in a fully factorised form.

3

1. Find the remainder when \(2 x ^ { 3 } - x ^ { 2 } + 2 x - 2\) is divided by \(2 x - 1\).
2. Given that \(\frac { 2 x ^ { 3 } - x ^ { 2 } + 2 x - 2 } { 2 x - 1 } = x ^ { 2 } + a + \frac { b } { 2 x - 1 }\), find the values of \(a\) and \(b\).

1

1. The polynomial \(\mathrm { p } ( x )\) is defined by \(\mathrm { p } ( x ) = 6 x ^ { 3 } - 19 x ^ { 2 } + 9 x + 10\).
   1. Find \(\mathrm { p } ( 2 )\).
   2. Use the Factor Theorem to show that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\).
   3. Write \(\mathrm { p } ( x )\) as the product of three linear factors.
2. Hence simplify \(\frac { 3 x ^ { 2 } - 6 x } { 6 x ^ { 3 } - 19 x ^ { 2 } + 9 x + 10 }\).

1

1. Find the remainder when \(2 x ^ { 2 } + x - 3\) is divided by \(2 x + 1\).
   (2 marks)
2. Simplify the algebraic fraction \(\frac { 2 x ^ { 2 } + x - 3 } { x ^ { 2 } - 1 }\).
   (3 marks)

1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 27 x ^ { 3 } - 9 x + 2\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(3 x + 1\).
2. 1. Show that f \(\left( - \frac { 2 } { 3 } \right) = 0\).
   2. Express \(\mathrm { f } ( x )\) as a product of three linear factors.
   3. Simplify $$\frac { 27 x ^ { 3 } - 9 x + 2 } { 9 x ^ { 2 } + 3 x - 2 }$$

1

1. Use the Remainder Theorem to find the remainder when \(3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5\) is divided by \(3 x - 1\).
2. Express \(\frac { 3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5 } { 3 x - 1 }\) in the form \(a x ^ { 2 } + b x + \frac { c } { 3 x - 1 }\), where \(a , b\) and \(c\) are integers.

7 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows the curve $$y = x ^ { 3 } - x + 1$$ The points \(A\) and \(B\) on the curve have \(x\)-coordinates - 1 and \(- 1 + h\) respectively. \includegraphics[max width=\textwidth, alt={}, center]{a0a30197-ca11-40d9-9ccd-30281c5e0fb4-05_978_1184_676_411}

1. 1. Show that the \(y\)-coordinate of the point \(B\) is $$1 + 2 h - 3 h ^ { 2 } + h ^ { 3 }$$
   2. Find the gradient of the chord \(A B\) in the form $$p + q h + r h ^ { 2 }$$ where \(p , q\) and \(r\) are integers.
   3. Explain how your answer to part (a)(ii) can be used to find the gradient of the tangent to the curve at \(A\). State the value of this gradient.
2. The equation \(x ^ { 3 } - x + 1 = 0\) has one real root, \(\alpha\).
   1. Taking \(x _ { 1 } = - 1\) as a first approximation to \(\alpha\), use the Newton-Raphson method to find a second approximation, \(x _ { 2 }\), to \(\alpha\).
   2. On Figure 1, draw a straight line to illustrate the Newton-Raphson method as used in part (b)(i). Show the points \(\left( x _ { 2 } , 0 \right)\) and \(( \alpha , 0 )\) on your diagram.

5 Kaya is investigating the function $$f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } - 12 x + 45$$ Kaya makes two statements.
Statement 1: \(\mathrm { f } ( 3 ) = 0\) Statement 2: this shows that ( \(x + 3\) ) must be a factor of \(\mathrm { f } ( x )\).
5

1. State, with a reason, whether each of Kaya's statements is correct. Statement 1: \(\_\_\_\_\) Statement 2: \(\_\_\_\_\) 5
2. Fully factorise f (x).

12

1. Prove that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\) 12
2. Factorise \(\mathrm { p } ( x )\) completely.
   12
3. Prove that there are no real solutions to the equation $$\frac { 30 \sec ^ { 2 } x + 2 \cos x } { 7 } = \sec x + 1$$

13

1. Given that $$P ( x ) = 125 x ^ { 3 } + 150 x ^ { 2 } + 55 x + 6$$ use the factor theorem to prove that ( \(5 x + 1\) ) is a factor of \(\mathrm { P } ( x )\).
   [0pt] [2 marks]
   13
2. Factorise \(\mathrm { P } ( x )\) completely.
   13
3. Hence, prove that \(250 n ^ { 3 } + 300 n ^ { 2 } + 110 n + 12\) is a multiple of 12 when \(n\) is a positive whole number.

11 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + ( b + 2 ) x ^ { 2 } + 2 ( b + 2 ) x + 8$$ where \(b\) is a constant.
11

1. Use the factor theorem to prove that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) for all values of \(b\).
   11
2. The graph of \(y = \mathrm { p } ( x )\) meets the \(x\)-axis at exactly two points.
   11 (b) (i) Sketch a possible graph of \(y = \mathrm { p } ( x )\) \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-20_1084_965_1619_532} 11 (b) (ii) Given \(\mathrm { p } ( x )\) can be written as $$\mathrm { p } ( x ) = ( x + 2 ) \left( x ^ { 2 } + b x + 4 \right)$$ find the value of \(b\). Fully justify your answer.

4

1. Express \(z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80\) as the product of two quadratic factors with real coefficients.
   [4 marks]
   4 It is given that \(1 - 3 \mathrm { i }\) is one root of the quartic equation
   堛的 增
   4
2. Find the value of \(p\) and the value of \(r\).

13

1. Two of the solutions to the equation \(\cos 6 \theta = 0\) are \(\theta = \frac { \pi } { 4 }\) and \(\theta = \frac { 3 \pi } { 4 }\) Find the other solutions to the equation \(\cos 6 \theta = 0\) for \(0 \leq \theta \leq \pi\) 13
2. Use de Moivre's theorem to show that $$\cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$ 13
3. Use the fact that \(\theta = \frac { \pi } { 4 }\) and \(\theta = \frac { 3 \pi } { 4 }\) are solutions to the equation \(\cos 6 \theta = 0\) to find a factor of \(32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1\) in the form ( \(a \cos ^ { 2 } \theta + b\) ), where \(a\) and \(b\) are integers.
   [0pt] [4 marks]
4. Hence show that $$\cos \left( \frac { 11 \pi } { 12 } \right) = - \sqrt { \frac { 2 + \sqrt { 3 } } { 4 } }$$ \includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-25_2492_1721_217_150}

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) Given that \(C _ { 1 }\)

• has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) is a quadratic function
• cuts the \(x\)-axis at the origin and at \(x = 4\)
• has a minimum turning point at ( \(2 , - 4.8\) )
  1. find \(\mathrm { f } ( x )\)

Given that \(C _ { 2 }\)

The curves \(C _ { 1 }\) and \(C _ { 2 }\) meet in the first quadrant at the point \(P\), shown in Figure 1.

Use algebra to find the coordinates of \(P\).

9. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( x + 5 ) \left( 3 x ^ { 2 } - 4 x + 20 \right)$$

1. Deduce the range of values of \(x\) for which \(\mathrm { f } ( x ) \geqslant 0\)
2. Find \(\mathrm { f } ^ { \prime } ( x )\) giving your answer in simplest form. The point \(R ( - 4,84 )\) lies on \(C\).
   Given that the tangent to \(C\) at the point \(P\) is parallel to the tangent to \(C\) at the point \(R\) (c) find the \(x\) coordinate of \(P\).
   (d) Find the point to which \(R\) is transformed when the curve with equation \(y = \mathrm { f } ( x )\) is transformed to the curve with equation,
   1. \(y = \mathrm { f } ( x - 3 )\)
   2. \(y = 4 \mathrm { f } ( x )\)

3. $$f ( x ) = 2 x ^ { 3 } - x ^ { 2 } + A x + B$$ where \(A\) and \(B\) are integers.
Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 3 )\) the remainder is 55

1. show that $$3 A - B = - 118$$ Given also that \(( 2 x - 5 )\) is a factor of \(\mathrm { f } ( x )\),
2. find the value of \(A\) and the value of \(B\).
3. Hence find the quotient when \(\mathrm { f } ( x )\) is divided by ( \(x - 7\) )

2 Let \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15\).

1. Show that \(\mathrm { f } ( 1 ) = 0\) and hence factorise \(x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15\) completely.
2. Hence solve the equation \(x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15 = 0\).

2

1. Show that \(x = 2\) is a root of the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
2. Hence solve the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).

5 When \(x ^ { 4 } - 4 x ^ { 3 } + 5 x ^ { 2 } + x + a\) is divided by \(x ^ { 2 } - x + 1\), the quotient is \(x ^ { 2 } + b x + 1\) and the remainder is \(c x - 3\). Find the values of the constants \(a , b\) and \(c\).

13 By first factorising completely \(x ^ { 3 } + x ^ { 2 } - 5 x + 3\), find \(\int \frac { 2 x ^ { 2 } + x + 1 } { x ^ { 3 } + x ^ { 2 } - 5 x + 3 } \mathrm {~d} x\).

10 Let \(\mathrm { f } ( x ) = x ^ { 4 } - 4 x ^ { 3 } - 10 x ^ { 2 } + 28 x - 15\).

1. Show that \(x = 1\) is a root of the equation \(\mathrm { f } ( x ) = 0\).
2. Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(x - 5\).
3. Factorise \(\mathrm { f } ( x )\) fully and hence sketch the graph of \(y = \mathrm { f } ( x )\).

10

1. Show that \(\sin \left( 2 \theta + \frac { 1 } { 2 } \pi \right) = \cos 2 \theta\).
2. Hence solve the equation \(\sin 3 \theta = \cos 2 \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
3. Show that \(\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta\). Hence, by writing \(\cos 2 \theta - \sin 3 \theta\) in terms of \(\sin \theta\), use your answer to part (ii) to determine the solutions of \(4 x ^ { 3 } - 2 x ^ { 2 } - 3 x + 1 = 0\).

4

1. Show that \(x = 2\) is a root of the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
2. Hence solve the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).

4

1. Show that \(x = 2\) is a root of the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
2. Hence solve the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).